Operating with external arguments in the Mandelbrot set antenna. (English) Zbl 1008.37028

Pastor, G.; Romera, M.; Álvarez, G.; Montoya, F.

Physica D 171, No. 1-2, 52-71 (2002).

Summary: The external argument theory of Douady and Hubbard allows us to know both the potential and the field-lines in the exterior of the Mandelbrot set. Nonetheless, there are no explicit formulae to operate with external arguments, and the external argument theory is difficult to apply. In this paper we introduce some tools in order to obtain formulae to operate with external arguments in the Mandelbrot set antenna. Thus, we introduce the harmonic tool to calculate both the external arguments of the period-doubling cascade hyperbolic components and the external arguments of the last appearance hyperbolic components. Likewise, we introduce composition rules applied to external arguments that, with the aid of the concept of heredity, allows the calculation of all the external arguments that constitutes the family tree of a given external argument.

MSC:

37F45

Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)

Keywords:

periodic-doubling cascade hyperbolic components; heredity concept; family tree

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